Quasi-maximum Likelihood Estimation and Inference in Dynamic Models with Time-varying Covariances

We study the properties of the quasi-maximum likelihood estimator (QMLE) and related test statistics in dynamic models that jointly parameterize conditional means and conditional covariances, when a normal log-likelihood is maximized but the assumption of normality is violated. Because the score of the normal log-likelihood has the martingale difference property when the first two conditional moments are correctly specified, the QMLE is generally consistent and has a limiting normal distribution. We provide easily computable formulas for asymptotic standard errors that are valid under nonnormality. Further, we show how robust LM tests for the adequacy of the jointly parameterized mean and variance can be computed from simple auxiliary regressions. An appealing feature of these robust inference procedures is that only first derivatives of the conditional mean and variance functions are needed. A Monte Carlo study indicates that the asymptotic results carry over to finite samples. Estimation of several AR and AR-GARCH time series models reveals that in most situations the robust test statistics compare favorably to the two standard (nonrobust) formulations of the Wald and LM tests. Also, for the GARCH models and the sample sizes analyzed here, the bias in the QMLE appears to be relatively small. An empirical application to stock return volatility illustrates the potential importance of computing robust statistics in practice.